Sr Examen
Integral de sqrt(1-(y-1)^2) dy
Solución
Ha introducido
[src]
2 / | | ______________ | / 2 | \/ 1 - (y - 1) dy | / 1
$$\int\limits_{1}^{2} \sqrt{1 - \left(y - 1\right)^{2}}\, dy$$
Integral(sqrt(1 - (y - 1)^2), (y, 1, 2))
Respuesta (Indefinida)
[src]
// 3 \
|| I*acosh(-1 + y) I*(-1 + y) I*(-1 + y) | 2| |
/ ||- --------------- + --------------------- - --------------------- for |(-1 + y) | > 1|
| || 2 ________________ ________________ |
| ______________ || / 2 / 2 |
| / 2 || 2*\/ -1 + (-1 + y) 2*\/ -1 + (-1 + y) |
| \/ 1 - (y - 1) dy = C + |< |
| || _______________ |
/ || / 2 |
|| asin(-1 + y) \/ 1 - (-1 + y) *(-1 + y) |
|| ------------ + --------------------------- otherwise |
|| 2 2 |
\\ /
$$\int \sqrt{1 - \left(y - 1\right)^{2}}\, dy = C + \begin{cases} \frac{i \left(y - 1\right)^{3}}{2 \sqrt{\left(y - 1\right)^{2} - 1}} - \frac{i \left(y - 1\right)}{2 \sqrt{\left(y - 1\right)^{2} - 1}} - \frac{i \operatorname{acosh}{\left(y - 1 \right)}}{2} & \text{for}\: \left|{\left(y - 1\right)^{2}}\right| > 1 \\\frac{\sqrt{1 - \left(y - 1\right)^{2}} \left(y - 1\right)}{2} + \frac{\operatorname{asin}{\left(y - 1 \right)}}{2} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta
[src]
2 / | | / 2 3 | | I 3*I*(-1 + y) I*(-1 + y) *(1 - y) I*(1 - y)*(-1 + y) 2 | |- ------------------- + --------------------- + --------------------- - --------------------- for (-1 + y) > 1 | | ________________ ________________ 3/2 3/2 | | / 2 / 2 / 2\ / 2\ | | \/ -1 + (-1 + y) 2*\/ -1 + (-1 + y) 2*\-1 + (-1 + y) / 2*\-1 + (-1 + y) / | | | < _______________ dy | | / 2 | | \/ 1 - (-1 + y) 1 (1 - y)*(-1 + y) | | ------------------ + -------------------- + -------------------- otherwise | | 2 _______________ _______________ | | / 2 / 2 | | 2*\/ 1 - (-1 + y) 2*\/ 1 - (-1 + y) | \ | / 1
$$\int\limits_{1}^{2} \begin{cases} \frac{i \left(1 - y\right) \left(y - 1\right)^{3}}{2 \left(\left(y - 1\right)^{2} - 1\right)^{\frac{3}{2}}} - \frac{i \left(1 - y\right) \left(y - 1\right)}{2 \left(\left(y - 1\right)^{2} - 1\right)^{\frac{3}{2}}} + \frac{3 i \left(y - 1\right)^{2}}{2 \sqrt{\left(y - 1\right)^{2} - 1}} - \frac{i}{\sqrt{\left(y - 1\right)^{2} - 1}} & \text{for}\: \left(y - 1\right)^{2} > 1 \\\frac{\left(1 - y\right) \left(y - 1\right)}{2 \sqrt{1 - \left(y - 1\right)^{2}}} + \frac{\sqrt{1 - \left(y - 1\right)^{2}}}{2} + \frac{1}{2 \sqrt{1 - \left(y - 1\right)^{2}}} & \text{otherwise} \end{cases}\, dy$$
=
=
2 / | | / 2 3 | | I 3*I*(-1 + y) I*(-1 + y) *(1 - y) I*(1 - y)*(-1 + y) 2 | |- ------------------- + --------------------- + --------------------- - --------------------- for (-1 + y) > 1 | | ________________ ________________ 3/2 3/2 | | / 2 / 2 / 2\ / 2\ | | \/ -1 + (-1 + y) 2*\/ -1 + (-1 + y) 2*\-1 + (-1 + y) / 2*\-1 + (-1 + y) / | | | < _______________ dy | | / 2 | | \/ 1 - (-1 + y) 1 (1 - y)*(-1 + y) | | ------------------ + -------------------- + -------------------- otherwise | | 2 _______________ _______________ | | / 2 / 2 | | 2*\/ 1 - (-1 + y) 2*\/ 1 - (-1 + y) | \ | / 1
$$\int\limits_{1}^{2} \begin{cases} \frac{i \left(1 - y\right) \left(y - 1\right)^{3}}{2 \left(\left(y - 1\right)^{2} - 1\right)^{\frac{3}{2}}} - \frac{i \left(1 - y\right) \left(y - 1\right)}{2 \left(\left(y - 1\right)^{2} - 1\right)^{\frac{3}{2}}} + \frac{3 i \left(y - 1\right)^{2}}{2 \sqrt{\left(y - 1\right)^{2} - 1}} - \frac{i}{\sqrt{\left(y - 1\right)^{2} - 1}} & \text{for}\: \left(y - 1\right)^{2} > 1 \\\frac{\left(1 - y\right) \left(y - 1\right)}{2 \sqrt{1 - \left(y - 1\right)^{2}}} + \frac{\sqrt{1 - \left(y - 1\right)^{2}}}{2} + \frac{1}{2 \sqrt{1 - \left(y - 1\right)^{2}}} & \text{otherwise} \end{cases}\, dy$$
Integral(Piecewise((-i/sqrt(-1 + (-1 + y)^2) + 3*i*(-1 + y)^2/(2*sqrt(-1 + (-1 + y)^2)) + i*(-1 + y)^3*(1 - y)/(2*(-1 + (-1 + y)^2)^(3/2)) - i*(1 - y)*(-1 + y)/(2*(-1 + (-1 + y)^2)^(3/2)), (-1 + y)^2 > 1), (sqrt(1 - (-1 + y)^2)/2 + 1/(2*sqrt(1 - (-1 + y)^2)) + (1 - y)*(-1 + y)/(2*sqrt(1 - (-1 + y)^2)), True)), (y, 1, 2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.